Issue 3, Volume 44, September 2019

2019

1. Frobenius and Cartier Algebras of Stanley–Reisner Rings (II) Alberto F. Boix , Santiago Zarzuela PDF

   ▸ Abstract

   It is known that the Frobenius algebra of the injective hull of the residue field of a complete Stanley–Reisner ring (i.e., a formal power series ring modulo a squarefree monomial ideal) can be only principally generated or infinitely generated as algebra over its degree zero piece, and that this fact can be read off in the corresponding simplicial complex; in the infinite case, we exhibit a 1–1 correspondence between potential new generators appearing on each graded piece and certain pairs of faces of such a simplicial complex, and we use it to provide an alternative proof of the fact that these Frobenius algebras can only be either principally generated or infinitely generated.

   571-586
2. Critical Paired Dominating Sets and Irreducible Decompositions of Powers of Edge Ideals Nguyen Thi Dung , Nguyen Thi Thanh Tam , Hoang Le Truong , Hoang Ngoc Yen PDF

   ▸ Abstract

   Let $G$ be a finite simple graph. A set $S$ of vertices is a critical paired dominating set of $G$, if every vertex is adjacent to a vertex in $S$ and the removal of any vertex does not change the matching number of $G$. In this paper, we give a characterization of graphs $G$ which has a critical paired dominating set in terms of the irreducible decomposition of powers of the edge ideal associated to $G$.

   587-601
3. Jacobi-Trudi Type Formula for Character of Irreducible Representations of $\frak {gl}(m|1)$ Nguyên Luong Thái Bình , Nguyên Thi Phuong Dung , Phùng Hô Hai PDF

   ▸ Abstract

   We prove a determinantal type formula to compute the irreducible characters of the general Lie superalgebra $\mathfrak {gl}(m|1)$ in terms of the characters of the symmetric powers of the fundamental representation and their duals. This formula was conjectured by $J$. van der Jeugt and E. Moens for the Lie superalgebra $\mathfrak {gl}(m|n)$ and generalizes the well-known Jacobi-Trudi formula.

   603-615
4. Homological Invariants of Powers of Fiber Products Hop D. Nguyen , Thanh Vu PDF

   ▸ Abstract

   Let $R$ and $S$ be polynomial rings of positive dimensions over a field $k$. Let $I \subseteq R$, $J \subseteq S$ be non-zero homogeneous ideals none of which contains a linear form. Denote by $F$ the fiber product of $I$ and $J$ in $T = R \otimes_k S$. We compute homological invariants of the powers of $F$ using the data of $I$ and $J$. Under the assumption that either $\mathrm{char\,} k = 0$ or $I$ and $J$ are monomial ideals, we provide explicit formulas for the depth and regularity of powers of $F$. In particular, we establish for all $s \geq  2$ the intriguing formula $\mathrm{depth}(T/F^s) = 0$. If moreover each of the ideals $I$ and $J$ is generated in a single degree, we show that for all $s \geq  1$, $\mathrm{reg\,} F^s = \max_{i\in [1, s]}\{\mathrm{reg\,} I^i + s − i, \mathrm{reg\,} J^i + s − i\}$. Finally, we prove that the linearity defect of $F$ is the maximum of the linearity defects of $I$ and $J$, extending previous work of Conca and Römer. The proofs exploit the so-called Betti splittings of powers of a fiber product.

   617-638
5. Inertial Extragradient Algorithms for Solving Equilibrium Problems Nguyen The Vinh , Le Dung Muu PDF

   ▸ Abstract

   We introduce two new algorithms based upon the extragradient and inertial methods for solving pseudomonotone equilibrium problems in real Hilbert spaces. Strong converge and weak convergence of the proposed algorithms are established under some mild assumptions. Numerical results show that the proposed algorithms are more efficient than some existing methods for equilibrium problems.

   639-663
6. Primary Decomposition of Powers of the Prime Ideal of a Numerical Semigroup Ring Ralf Fröberg PDF

   ▸ Abstract

   Let $R=k[t^{n_{1}},\ldots ,t^{n_{s}}]=k[x_{1},\ldots ,x_{s}]/P$ be a numerical semigroup ring and let $P^{(n)} = P^nR_P \cap R$ be the symbolic power of $P$ and $R_s(P) = \oplus_{i\geq 0}P^{(n)}t^n$ the symbolic Rees ring of $P$. It is hard to determine symbolic powers of $P$; there are even non-Noetherian symbolic Rees rings for 3-generated semigroups. We determine the primary decomposition of powers of $P$ for some classes of 3-generated numerical semigroups.

   665-669
7. Eakin-Sathaye-Type Theorems for Joint Reductions and Good Filtrations of Ideals Kriti Goel , Sudeshna Roy , J. K. Verma PDF

   ▸ Abstract

   Analogues of Eakin-Sathaye theorem for reductions of ideals are proved for $\mathbb N^s$-graded good filtrations. These analogues yield bounds on joint reduction vectors for a family of ideals and reduction numbers for $\mathbb N$-graded filtrations. Several examples related to lex-segment ideals, contracted ideals in 2-dimensional regular local rings and the filtration of integral and tight closures of powers of ideals in hypersurface rings are constructed to show effectiveness of these bounds.

   671-689
8. 4-Dimensional Licci Gorenstein Stanley-Reisner Ideals Giancarlo Rinaldo , Naoki Terai PDF

   ▸ Abstract

   We classify licci Gorenstein squarefree monomial ideals $I$ with $\dim S/I \geq  4$, where S is a polynomial ring.

   691-700
9. A Bohr-Nikol’skii Inequality for Weighted Lebesgue Spaces Ha Huy Bang , Vu Nhat Huy PDF

   ▸ Abstract

   In this paper, we give a new inequality for weighted Lebesgue spaces called Bohr-Nikol’skii inequality, which combines the inequality of Bohr-Favard and the Nikol’skii idea of inequality for functions in different metrics.

   701-710
10. On Topologized Fundamental Groups with Small Loop Transfer Viewpoints Noorollah Jamali , Behrooz Mashayekhy , Hamid Torabi , Seyyed Zeynal Pashaei , Mehdi Abdullahi Rashid PDF

    ▸ Abstract

    In this paper, by introducing some kind of small loop transfer spaces at a point, we study the behavior of topologized fundamental groups with the compact-open topology and the whisker topology, $\pi _{1}^{qtop}(X,x_{0})$ and $\pi _{1}^{wh}(X,x_{0})$, respectively. In particular, we give necessary or sufficient conditions for equality and being topological group of these two topologized fundamental groups. Finally, we give some examples to show that the converse of some of these implications do not hold, in general.

    711-722
11. Elliptic Solutions to Nonsymmetric Monge-Ampère Type Equations II. A Priori Estimates and the Dirichlet Problem Ha Tien Ngoan , Thai Thi Kim Chung PDF

    ▸ Abstract

    In this paper, we consider the Dirichlet problem for nonsymmetric Monge-Ampère type equations in which a skew-symmetric matrix is introduced. We establish uniform with respect to a class of skew-symmetric matrix bounds for $\delta$-elliptic $C^{2, \alpha }(\overline {{\Omega }})$-solutions to the Dirichlet problem. Then, we prove the classical solvability of the Dirichlet problem, provided those skew-symmetric matrices are sufficiently small in some sense.

    723-749
12. Absolutely $lq$-Finite Extensions El Hassane Fliouet PDF

    ▸ Abstract

    We describe the lower quasi-finite extensions $K/k$ of characteristic $p > 0$, which are defined as follows: for every $n\in \mathbb N, k^{p^{-n}} \cap K/k$ is finite. We are especially interested in examining the absolute case. In this regard, we give necessary and sufficient condition for an absolutely lq-finite extension to be of finite size. Moreover, we show that any extension that is at the same time modular and $lq$-finite is of finite size. Furthermore, we construct an example of extension $K/k$ of infinite size such that for any intermediate field $L$ of $K/k$ is of finite size over $k$.

    751-779
13. Sharp Constant for Poincaré-Type Inequalities in the Hyperbolic Space Quốc Anh Ngô , Van Hoang Nguyen PDF

    ▸ Abstract

    In this note, we establish a Poincaré-type inequality on the hyperbolic space $\mathbb H^n$, namely $$\|u\|_{p} \leq C(n,m,p) \|{\nabla^{m}_{g}} u\|_{p}$$ for any $u \in W^{m,p}(\mathbb {H}^{n})$. We prove that the sharp constant $C(n,m,p)$ for the above inequality is $$C(n,m,p) = \left\{\begin{array}{ll} \left( p p^{\prime}/(n-1)^{2} \right)^{m/2}&\text{if}~m~\text{is even},\\ (p/(n-1))\left( p p^{\prime}/(n-1)^{2}\right)^{(m-1)/2} &\text{if}~m~\text{is odd}, \end{array}\right.$$ with $p^\prime = p/(p − 1)$ and this sharp constant is never achieved in $W^{m,p}(\mathbb {H}^{n})$. Our proofs rely on the symmetrization method extended to hyperbolic spaces.

    781-795
14. Arithmetic Properties of 9-Regular Partitions with Distinct Odd Parts B. Hemanthkumar , H. S. Sumanth Bharadwaj , M. S. Mahadeva Naika PDF

    ▸ Abstract

    Let $\text{pod}_{9}(n)$, $\text{ped}_{9}(n)$, and $\overline{A}_{9}(n)$ denote the number of 9-regular partitions of n wherein odd parts are distinct, even parts are distinct, and the number of 9-regular overpartitions of $n$, respectively. By considering $\text{pod}_{9}(n)$ from an arithmetic point of view, we establish a number of infinite families of congruences modulo 16 and 32, and some internal congruences modulo small powers of 3. A relation connecting above partition functions in arithmetic progressions is obtained as follows. For any $n\geq 0$, $ 6 \text{pod}_{9}(2n + 1) = 2 \text{ped}_{9}(2n + 3) = 3 \overline{A}_{9}(n + 1).$

    797-811